Problem: Factor the following expression: $4$ $x^2$ $-9$ $x+$ $2$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(4)}{(2)} &=& 8 \\ {a} + {b} &=& & & {-9} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $8$ and add them together. The factors that add up to ${-9}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-1}$ and ${b}$ is ${-8}$ $ \begin{eqnarray} {ab} &=& ({-1})({-8}) &=& 8 \\ {a} + {b} &=& {-1} + {-8} &=& -9 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {4}x^2 {-1}x {-8}x +{2} $ Group the terms so that there is a common factor in each group: $ ({4}x^2 {-1}x) + ({-8}x +{2}) $ Factor out the common factors: $ x(4x - 1) - 2(4x - 1) $ Notice how $(4x - 1)$ has become a common factor. Factor this out to find the answer. $(4x - 1)(x - 2)$